Calculus: Master Differential and Integral Calculus Concepts

Question 1

Find the integral of the function f(x) = e^(3x).

Accepted Answer

(1/3)e^(3x) + C

Answer

e^(3x) + C

Answer

(3/1)e^(3x) + C

Answer

3e^(3x) + C

Question 2

Which of the following is NOT a theorem in Calculus?

Accepted Answer

Schrödinger's Cat Theorem

Answer

Mean Value Theorem

Answer

Chain Rule

Answer

Taylor's Theorem

Question 3

The derivative of a constant function is:

Accepted Answer

0

Answer

1

Answer

x

Answer

undefined

Question 4

Find the derivative of the polynomial function f(x) = x^3 + 2x^2 - 5x + 1.

Accepted Answer

3x^2 + 4x - 5

Answer

x^2 + 4x - 5

Answer

3x^2 - 5

Answer

x^2 - 5

Question 5

The definite integral ∫[0, 2] x^2 dx represents the area under the curve of the function f(x) = x^2 from x = 0 to x = 2.

Accepted Answer

True

Answer

False

Question 6

The volume of the solid generated by rotating the region bounded by the curves y = x^2 and y = 4 about the x-axis is given by π∫[0, 2] (4^2 - x^4) dx.

Accepted Answer

True

Answer

False

Question 7

The work done by the force F(x) = x^2 + 1 acting on an object that moves from x = 0 to x = 2 is given by ∫[0, 2] (x^2 + 1) dx.

Accepted Answer

True

Answer

False

Question 8

The area of the region bounded by the curves y = x^2 and y = 2x + 1 is given by ∫[0, 1] (2x + 1 - x^2) dx.

Accepted Answer

True

Answer

False

Question 9

The length of the curve y = x^3/3 from x = 0 to x = 1 is given by ∫[0, 1] (1 + x^2) ^ (1/2) dx.

Accepted Answer

True

Answer

False

Question 10

What is the order of the integral ∫∫∫x^3 y^2 z dz dy dx?

Accepted Answer

4

Answer

1

Answer

2

Answer

3

Question 11

Find the derivative of the function f(x) = x³ + 2x² - 5x + 1.

Accepted Answer

3x² + 4x - 5

Answer

x² + 2x - 5

Answer

3x² + 2

Answer

x³

Question 12

What is the limit of (x² - 4)/(x - 2) as x approaches 2?

Accepted Answer

0

Answer

2

Answer

4

Answer

The limit does not exist

Question 13

What is the derivative of the function f(x) = sin(x)?

Accepted Answer

cos(x)

Answer

-cos(x)

Answer

sin(x)

Answer

-sin(x)

Question 14

Which of the following is the derivative of the function f(x) = ln(x)?

Accepted Answer

1/x

Answer

-1/x

Answer

x

Question 15

Find the derivative of the function (f(x) = e^(2x) + ln(x^3)).

Accepted Answer

2e^(2x) + 3x^(2)

Answer

2e^(2x) + x^(-2)

Answer

2e^(2x) + 3/x

Answer

e^(2x) + 3x^(-2)

Question 16

Use the Intermediate Value Theorem to determine if the function (f(x) = x^3 - 2x^2 + 1) has a root in the interval [0, 1].

Accepted Answer

Yes, the function has a root in the interval.

Answer

No, the function does not have a root in the interval.

Answer

The Intermediate Value Theorem cannot be applied to this function.

Answer

The information given is not sufficient to determine if the function has a root in the interval.

Question 17

Which of the following techniques can be used to evaluate the integral of the function (f(x) = (x^2 + 1) / (x^3 + 3x))?

Accepted Answer

Partial fractions

Answer

Integration by substitution

Answer

Integration by parts

Answer

Trig substitution

Question 18

Which of the following is the formal definition of the limit of a function f(x) as x approaches a?

Accepted Answer

For any positive number ε, there exists a positive number δ such that if 0 < |x - a| < δ, then |f(x) - L| < ε

Answer

For any positive number δ, there exists a positive number ε such that if 0 < |f(x) - L| < ε, then |x - a| < δ

Answer

For any number L, there exists a positive number δ such that if 0 < |x - a| < δ, then |f(x) - L| < ε

Answer

For any number ε, there exists a positive number δ such that if |x - a| > δ, then |f(x) - L| < ε

Question 19

Find the derivative of the function f(x) = x^2 + 2x - 1.

Accepted Answer

2x + 2

Answer

x^2 + 2x

Answer

1

Answer

x

Question 20

Find the integral of the function f(x) = sin(x).

Accepted Answer

cos(x) + C

Answer

sin(x) + C

Answer

-cos(x) + C

Answer

xcos(x) + C

Question 21

Find the second derivative of the function f(x) = sin(2x).

Accepted Answer

-4sin(2x)

Answer

-2sin(2x)

Answer

2sin(2x)

Answer

4sin(2x)

Question 22

Determine the derivative of the logarithmic function f(x) = ln(x).

Accepted Answer

1/x

Answer

-1/x

Answer

x

Answer

-x

Question 23

State the formula for the product rule.

Accepted Answer

f'(x)g(x) + f(x)g'(x)

Answer

f'(x)g'(x)

Answer

f(x)g(x)

Answer

f(x)/g(x)

Question 24

Calculate the integral of the function f(x) = eˣ.

Accepted Answer

eˣ + C

Answer

eˣ

Answer

x*eˣ

Answer

x*eˣ + C

Question 25

Determine the indefinite integral of the function f(x) = 1 / (x² + 1).

Accepted Answer

arctan(x) + C

Answer

tan(x) + C

Answer

ln(x² + 1) + C

Answer

x² + C

Question 26

To find the derivative of the function f(x) = x^2, we apply the power rule, which states that the derivative of x^n is:

Accepted Answer

nx^(n-1)

Answer

x^(n+1)

Answer

1/x^n

Answer

x^n

Question 27

Which of the following is NOT a technique for evaluating definite integrals?

Accepted Answer

Using a graphing calculator

Answer

The Fundamental Theorem of Calculus

Answer

Integration by substitution

Answer

Integration by parts

Question 28

Determine if the function f(x) = x^2 + 2x - 3 is increasing or decreasing on the interval (-∞, -1) by analyzing the sign of its first derivative.

Accepted Answer

Decreasing

Answer

Constant

Answer

Increasing

Question 29

Determine the convergence or divergence of the series ∑(n=1 to ∞) 1 / n^2 using the integral test.

Accepted Answer

Convergent

Answer

Oscillating

Answer

Cannot be determined

Answer

Divergent

Question 30

Find the derivative of f(x) = x^3 - 2x^2 + 5 using the power rule.

Accepted Answer

3x^2 - 4x

Answer

x^3 - 4x^2

Answer

x^3 - 4x + 5

Answer

3x^3 - 2x

Question 31

Find the integral of ∫(2x + 1) dx using the power rule.

Accepted Answer

x^2 + x + C

Answer

x^2 + C

Answer

2x^2 + x + C

Answer

x^2 - x + C

Question 32

Find the maximum value of the function f(x) = -x^2 + 4x + 3 using the first derivative test.

Accepted Answer

5

Answer

3

Answer

7

Answer

-5

Question 33

Which of the following differential equations represents exponential growth?

Accepted Answer

dy/dx = ky

Answer

dy/dx = x

Answer

dy/dx = y^2

Answer

d^2y/dx^2 = y

Question 34

Find the derivative of the function:
$f(x) = x^3 - 2x^2 + 5x - 1$

Accepted Answer

3x^2 - 4x + 5

Answer

x^3 - 4x^2 + 5x - 1

Answer

3x - 2

Answer

5x - 1

Question 35

Calculate the area under the curve y = 2x - x² between x = 0 and x = 2.

Accepted Answer

2/3

Answer

1

Answer

1/3

Question 36

Find the maximum value of the function f(x) = -x² + 4x + 3 on the interval [-1, 3].

Accepted Answer

7

Answer

-7

Answer

3

Answer

5

Question 37

Determine whether the function f(x) = |x - 2| is differentiable at x = 2.

Accepted Answer

No

Answer

Only from the left

Answer

Yes

Answer

Only from the right

Question 38

Find the volume of the solid generated by rotating the region bounded by the curves y = x² and y = 4 about the x-axis.

Accepted Answer

32π/5

Answer

16π/5

Answer

128π/5

Answer

64π/5

Question 39

What is the derivative of the function f(x) = x³?

Accepted Answer

3x²

Answer

⅓x²

Answer

2x³

Answer

x²

Question 40

Calculate the limit of (x² - 1) / (x - 1) as x approaches 1.

Accepted Answer

2

Answer

1

Answer

0

Answer

Does not exist

Question 41

Which definite integral represents the area under the curve y = 2x from x = 0 to x = 2?

Accepted Answer

∫[0,2] 2x dx

Answer

∫[0,2] x² dx

Answer

∫[0,2] 2 dx

Answer

2 ∫[0,2] x dx

Question 42

What is the slope of the line tangent to the curve y = sin(x) at the point where x = π/4?

Accepted Answer

√2 / 2

Answer

1

Answer

0

Answer

-√2 / 2

Question 43

Determine the indefinite integral of e^(2x) dx.

Accepted Answer

(1/2) * e^(2x) + C

Answer

e^(2x) + C

Answer

2 * e^(2x) + C

Answer

This function cannot be integrated

Question 44

Which of the following functions has an inverse function?

Accepted Answer

f(x) = x³

Answer

f(x) = sin(x)

Answer

f(x) = x²

Answer

f(x) = |x|

Question 45

Which of the following expressions defines the derivative of a function f(x)?

Accepted Answer

f'(x) = lim_(h→0) [f(x+h) - f(x)]/h

Answer

f'(x) = [f(x) - f(x+h)]/h

Answer

f'(x) = f(x) + f(x+h)

Answer

f'(x) = f(x) - f(x+h)

Question 46

Compute the derivative of the function f(x) = x^3 - 5x^2 + 2x.

Accepted Answer

f'(x) = 3x^2 - 10x + 2

Answer

f'(x) = x^3 - 5x^2 + 2

Answer

f'(x) = 3x^2 - 10x - 2

Answer

f'(x) = x^3 - 5x + 2

Question 47

Which of the following correctly describes the Chain Rule in differentiation?

Accepted Answer

If f(u) is differentiable and u = g(x) is differentiable, then the derivative of f(g(x)) with respect to x is f'(g(x)) * g'(x).

Answer

If f(u) is differentiable and u = g(x) is differentiable, then the derivative of f(g(x)) with respect to x is f'(x) - g'(x).

Answer

If f(u) is differentiable and u = g(x) is differentiable, then the derivative of f(g(x)) with respect to x is f'(x) * g'(x).

Answer

If f(u) is differentiable and u = g(x) is differentiable, then the derivative of f(g(x)) with respect to x is f'(g(x)) + g'(x).

Question 48

Find the derivative of the function f(x) = cos(3x - 1).

Accepted Answer

f'(x) = -3sin(3x - 1)

Answer

f'(x) = cos(3x - 1)

Answer

f'(x) = 3sin(3x - 1)

Answer

f'(x) = -sin(3x - 1)

Question 49

Which of the following is the Fundamental Theorem of Calculus, Part 1?

Accepted Answer

If f(x) is continuous on the interval [a, b], then the definite integral of f(x) from a to b exists and is equal to the net area between the graph of f(x) and the x-axis on the interval [a, b].

Answer

If f(x) is continuous on the interval [a, b], then the definite integral of f(x) from a to b does not exist.

Question 50

Find the general antiderivative of the function f(x) = 2x + e^x.

Accepted Answer

F(x) = x^2 + e^x + C

Answer

F(x) = x^2 - e^x + C

Answer

F(x) = 2x - e^x + C

Answer

F(x) = 2x^2 + e^x + C

Question 51

What is the derivative of the function f(x) = x^3 + 4x^2 - 2?

Accepted Answer

f'(x) = 3x^2 + 8x

Answer

f'(x) = x^3 + 8x^2

Answer

f'(x) = x^3 + 4

Answer

f'(x) = 3x^2 + 4x

Question 52

What is the limit of (x^2 - 4)/(x - 2) as x approaches 2?

Accepted Answer

4

Answer

2

Answer

0

Answer

Undefined

Question 53

Which of the following is the logarithmic derivative of the function f(x) = ln(x^2 + 1)?

Accepted Answer

2x/(x^2 + 1)

Answer

ln(x^2 + 1)

Answer

1/(x^2 + 1)

Answer

(x^2 + 1)/2x

Question 54

Find the inverse function of f(x) = (x - 1) / (x + 1).

Accepted Answer

f^-1(x) = (x + 1) / (x - 1)

Answer

-1 / (x + 1)

Answer

1 / (x - 1)

Answer

(x + 1) / (x - 1)

Question 55

Which of the following is the antiderivative of the function f(x) = 1 / x?

Accepted Answer

ln |x| + C

Answer

x + C

Answer

-1 / x^2 + C

Answer

1 / x + C

Question 56

Find the derivative of the function f(x) = 3x^2 + 2x - 1.

Accepted Answer

f'(x) = 6x + 2

Answer

f'(x) = 3x + 2

Answer

f'(x) = 6x^2 - 2

Answer

f'(x) = 9x^2 + 2

Question 57

Evaluate the limit of (x^2 - 4) / (x - 2) as x approaches 2.

Accepted Answer

4

Answer

Does not exist

Answer

0

Answer

2

Question 58

Find the equation of the tangent line to the curve y = x^3 at the point (1, 1).

Accepted Answer

y = 3x - 2

Answer

y = x - 2

Answer

y = 3x + 1

Answer

y = x + 1

Question 59

Calculate the integral of the function (2x + 1) dx.

Accepted Answer

x^2 + x + C

Answer

2x^2 + x + C

Answer

2x + C

Answer

x^2 + C

Question 60

Find the area under the curve y = x^2 from x = 0 to x = 2.

Accepted Answer

8/3

Answer

2

Answer

16/3

Answer

4

Question 61

Identify the critical points of the function f(x) = x^3 - 3x^2 + 2.

Accepted Answer

x = 0, x = 2

Answer

x = 1, x = 2

Answer

x = 1, x = 3

Answer

x = 0, x = 1

Question 62

Calculate the volume of the solid generated by revolving the region bounded by the curve y = √x, the x-axis, and the line x = 4 about the x-axis.

Accepted Answer

8π

Answer

4π

Answer

16π

Answer

32π

Question 63

Find the derivative of the function ln(x^2 + 1).

Accepted Answer

2x / (x^2 + 1)

Answer

(x^2 + 1) / 2x

Answer

2x

Answer

1 / (x^2 + 1)

Question 64

Calculate the derivative of the function f(x) = 3x² + 2x - 1.

Accepted Answer

6x + 2

Answer

3x²

Answer

3x + 2

Answer

9x